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GCD LCM Formula

In this lesson, I will formalize something that I suggested in the last lesson about the relationship between greatest common factor and least common multiple. Let’s begin with a numerical example, finding the greatest common factor and least common multiple of 18 and 24. First, we find the prime factorizations.

Of course 18 is 2 times 9. Break that into prime factors. 24 is 4 times 6. Break that into factors. Those are the two prime factorizations. The next step is to identify common factors, and they each have at least one factor of 2 and one factor of 3.

So 2 times 3, those are the common factors. The greatest common factor is the product of those, 2 times 3 equals 6. That is the greatest common factor of 18 and 24. Now, to construct the least common multiple, we could either use all the factors in 18, those three factors, in addition, the two extra factors in 24, the 2 times 2.

Or, we could use all the factors in 24, and in addition, all the extra factors of 18. Which would just be the extra 3. Either way, this is gonna lead to 2 times 2 times 2 times 3 times 3, which is 8 times 9, which is 72. So to make clear this choice, this is a very important choice, we had a choice either to do 24 times whatever 18 divided by the greatest common factor is, or we could do 18 times whatever 24 divided by the common factor is.

And either equals the least common multiple. Very interesting. We could consolidate that in the form least common multiple equals 18 x 24 over the greatest common factor. And this suggests a general pattern. More generally, we could say that for any two integers, P and Q, the least common multiple equals P times Q divided by the greatest common factor.

So this is the formula that very elegantly unites all these ideas. We have to be extremely careful with this formula, because this formula, even though it’s correct, invites a certain mathematical misunderstanding, a certain misconception about fractions. The dangers of this formula is that a student might be tempted to make the horrible mistake of performing the multiplication first.

More then most other contexts, it is very important here to cancel here before we do any multiplying. You see, the misconception that people have is when they see that multiplication in the numerator, they assume that it is one of God’s absolute commandments that they have to do that multiplication first.

And that is a complete misconception. That is a misconception that will get you in big trouble on the, on the test. So please do not fall for that misconception. That’s gonna lead to horribly big numbers that make your life much more difficult than it has to be. This is a formula that absolutely invites, do the cancellation fist.

Get the numbers smaller first. Divide either P by the denominator, or divide Q by the denominator, either way, to make your life easier. To show you an example, let’s find the least common multiple of 48 and 75. The greatest common factor is 3. So we’ll put this together.

The least common multiple, P times Q over greatest common factor, 48 times 75 equals 3. So, fraction misconception number one, of course it would be a huge, huge strategic mistake to perform the multiplication of 48 x 75 first. That would be a huge mistake. We want to perform the division by 3 first.

Fraction misconception number 2, is that some people think that we have to divide both the 48 and the 75 by 3. That is another fraction misconception. When we divide by 3, we only divide once. We can divide over the 48 or the 75, we do not divide both. And if you are having trouble with the rules of fractions here, I strongly suggest that you go back and watch some of the videos that discuss fractions.

So in this problem, I’m going to choose to divide the 75 by 3. That gets me down to 48 times 25. And the reason I did that is I can now use the doubling and halving trick. I’m gonna double 25, which is 50, and take half of 48, which is 24. So now, I have 24 times 50. I’m gonna double and halve again.

I’m gonna take double the 50, which is 100, and half of 24, which is 12. That gives me 12 times 100. Well that I can do without a calculator. 12 times 100, that’s 1200. That is the least common multiple of 48 and 75. So the formula, least common multiple equals P times Q over greatest common factor, can enormously simplify the calculations to find the least common multiple.

Once again, it is very important to have a crystal clear understanding of the laws of fractions so that you can use this equation properly.